On the number of divisors which are values of a polynomial

Let τ(n) be the number of positive divisors of an integer n, and for a polynomial P(X)∈ℤ[X], let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau_{P}(n)=\left\vert{\left\{{P(m)>0:\ m\in\mathbb{Z},P(m)\mid n}\right\}}\right\vert.$$\end{document} R. de la Bretèche studied the maximum values of τP(n) in intervals. Here the following is proved: if P(X)∈ℤ[X] is not of the form a(X+b)k with a,b∈ℚ, and k∈ℕ then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau_{P}(n)\ll(\log n)\tau(n)^{3/5}.$$\end{document} This improves partially on La Bretèche’s results.